MARKOV CONVEXITY AND NONEMBEDDABILITY OF THE HEISENBERG GROUP

作者：Li Sean^*

来源：Annales de l'Institut Fourier, 2016, 66(4): 1615-1651.

DOI：10.5802/aif.3045

• Heisenberg group
• Markov convexity
• biLipschitz
• embeddings

We show that the continuous infinite dimensional Heisenberg group H-infinity is Markov 4-convex and that the 3-dimensional Heisenberg group H-1 (and thus also H-infinity) cannot be Markov p-convex for any p < 4. As Markov convexity is biLipschitz invariant and Hilbert spaces are Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not biLipschitz embed into any Euclidean space.
The Markov convexity lower bound follows from exhibiting an explicit embedding of Laakso graphs G(n) into H-infinity that has distortion at most Cn(1/4)root log n. We use this to derive a quantitative lower bound for the biLipschitz distortion of balls of the discrete Heisenberg group into Markov p-convex metric spaces. Finally, we show surprisingly that Markov 4-convexity does not give the optimal distortion for embeddings of binary trees B-m into H-infinity by showing that the distortion is on the order of root log m.

• 出版日期2016